Hereditary classes defined by finitely many excluded induced subgraphs have bounded tree-α iff they are (tw,ω)-bounded, i.e., exclude K_{a,a}, forests with components of at most three leaves, and their line graphs.
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The threshold for G(n,p) arrow (mH)_2 is n^{-1/max{m2(H),1}} with m approximately n/(2k-alpha), matching the Rodl-Rucinski threshold for most H.
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Tree-alpha and excluding finitely many graphs
Hereditary classes defined by finitely many excluded induced subgraphs have bounded tree-α iff they are (tw,ω)-bounded, i.e., exclude K_{a,a}, forests with components of at most three leaves, and their line graphs.
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Ramsey properties for tilings in random graphs
The threshold for G(n,p) arrow (mH)_2 is n^{-1/max{m2(H),1}} with m approximately n/(2k-alpha), matching the Rodl-Rucinski threshold for most H.